Let 1 < p < ∞ andGp = {Ixalan{Xk}k≤N € l³ (R) : ΣXk=x² = 0k=1Show that(i) the set Gp is a subspace of ² (R);P(ii) for 1 < p <∞, the set Gp is not closed;(iii) for p = 1, the set Gp is closed.Hint: Gp is the preimage of {0} under a certain mapping.

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Answered: Let 1 < p < ∞ and Gp = {Ixalan {Xk}k≤N…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q. 8.(a) Let {e} be an orthonormal set in a Hilbert space H, then Σ|(x,e₁) | ² ≤||x||² for each vector x in H.
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Let 1 < p < ∞ and Gp = {Ixalan {Xk}k≤N € l³ (R) : ΣXk= x² = 0 k=1 Show that (i) the set Gp is a subspace of ² (R); P (ii) for 1 < p <∞, the set Gp is not closed; (iii) for p = 1, the set Gp is closed. Hint: Gp is the preimage of {0} under a certain mapping.